A circumradius-inradius identity and an identity relating volumes for a simplex (Q1381447)

The author establishes some new identities relating the circum- and inradius of an \(n\)-simplex \(S\) as well as the volume of \(S\) and its pedal volume (i.e., the volume of the pedal simplex \(S'\) of \(S\), which is the convex hull of the orthogonal projection images of a suitable point within the facets of \(S)\). It turns out that known inequalities by L. Fejes Tóth, M. S. Klamkin, L. Gerber and others are consequences of these relations, and even slight generalizations of these inequalities can be obtained.